An Analysis of Actin Delivery in the Acrosomal Process of Thyone

An Analysis of Actin Delivery in the Acrosomal Process of Thyone

Donald J. Olbris and Judith HerzfeldDept. of Chemistry and Keck Institute for Cellular Visualization, Brandeis University, Waltham, Massachusetts 02454-9110 USA

ABSTRACT The acrosomal process of the sea cucumber Thyone briareus can extend 90 �m in 10 s, but an epithelialgoldfish keratocyte can only glide a few microns in the same time. Both speeds reflect the rate of extension of an actinnetwork. The difference is in the delivery of actin monomers to the polymerization region. Diffusion supplies monomers fastenough to support the observed speed of goldfish keratocytes, but previous models have indicated that the acrosomalprocess of Thyone extends too rapidly for diffusion to keep up. Here we reexamine the assumptions made in earlier modelsand present a new model, the Actin Reconcentration Model, that includes more biological detail. Salt and water fluxes duringthe acrosomal reaction and the nonideality of the cytoplasm are particularly significant for actin delivery. We find that thevariability of the acrosomal growth curve can be explained by the salt and water fluxes, and that nonideality magnifies theeffect of actin concentration changes. We calculate the speed of process growth using biologically relevant parameters fromthe literature and find that the predictions of the model fall among the experimental data.

INTRODUCTION

Humans and other vertebrates do not grow new limbs towalk or reach. They simply move the limbs they alreadyhave using the attached muscles. Cells employ differenttactics. By controlled, localized polymerization of actinmonomers, cells create a variety of protrusions and exten-sions with which to crawl, reach, and glide. Cytoskeletalrearrangement is used for cell locomotion (Bray and White,1988; Cooper, 1991; Mitchison and Cramer, 1996) and cellprotrusion (Condeelis, 1993; Oster and Perelson, 1987),including fibroblast migration (Conrad et al., 1989; DeBia-sio, et al., 1988), neuronal growth cone extension (Smith,1988), and amoeba motility (Grebecki, 1994). Some patho-gens also use their host’s actin to move. Listeria monocy-togenes (Cossart, 1995; Southwick and Purich, 1994; Tilneyet al., 1992a, b), Shigella flexneri (Goldberg and Theriot,1995; Theriot, 1995), members of the Rickettsia family(Heinzen et al., 1993; Teysseire et al., 1992), and theVaccinia virus (Cudmore et al., 1995) initiate polymeriza-tion of actin into “comet tails” that propel them through thecytoplasm. Another pathogen, enteropathogenic Esche-richia coli, is propelled over the external surface of a cell by“pedestals” of actin that it induces to form inside the cell(Sanger et al., 1996).

Along with the wide structural variation in systems withactin-based motility, there is a wide range of observedspeeds (Condeelis, 1993). Although epithelial goldfish ker-atocytes and Listeria move at speeds of tenths of micronsper second (Lee et al., 1993; Theriot and Mitchison, 1991;Theriot et al., 1992), fibroblasts crawl relatively slowly, at�0.01 �m/s (Theriot and Mitchison, 1992). In contrast, theextension of the acrosomal process of the sea cucumber

Thyone briareus is rapid. When Thyone sperm contacts eggjelly, it constructs a thin extension with an actin core: theacrosomal process (Dan, 1967; Inoue and Tilney, 1982).The process grows rapidly and reaches lengths of 60–90�m in under 10 s (Colwin and Colwin, 1955). By compar-ison, a fast-moving goldfish keratocyte might take a fewminutes to crawl the same distance; a 3T3 fibroblast wouldtake hours (Theriot and Mitchison, 1992).

Like crawling cells, the acrosomal process polymerizesactin into filaments at its leading edge (Tilney, 1978; Tilneyand Kallenbach, 1979). It therefore requires the continuousdelivery of actin monomers to its tip in order to continueextending. The simplest delivery mechanism is diffusion.As polymerization consumes actin, a concentration gradientforms between the interior reservoir and the leading edge,causing actin monomers to diffuse down the gradient. Forcrawling goldfish keratocytes, monomers depolymerize atthe rear of the lamellipodium and diffuse forward to theleading edge; where they are rendered polymerization-com-petent. In this system, the diffusive flux is adequate tosupport the observed motility rate (Olbris and Herzfeld,1996).

Two features of the acrosomal process make diffusiveactin delivery less efficient than in the fish keratocyte. First,because Thyone’s acrosomal process reaches such extremelengths (up to 90 �m compared to the 3–5-�m lamellipo-dium of a goldfish keratocyte), actin must be transportedover much longer distances than in other systems. Diffusiveflux decreases inversely with increasing distance. Second,the density of actin filaments in the acrosomal process is fargreater than in the fish keratocyte lamellipodium. Not onlydoes the higher density of filaments require an increasedflux of actin monomers to maintain the extension speed, butthe space occupied by the filaments themselves also reducesthe fluid volume available for actin transport.

In any case, to properly explain the process’s extensionspeed, it is necessary to examine more closely what happensduring the whole acrosomal reaction, of which the extension

Received for publication 13 May 1999 and in final form 30 August 1999.Address reprint requests to Dr. Judith Herzfeld, Dept. of Chemistry, MS015, Brandeis University, 415 South St., Waltham, MA 02454-9110. Tel.:781-736-2538; Fax: 781-736-2516; E-mail: [email protected].

© 1999 by the Biophysical Society

0006-3495/99/12/3407/17 $2.00

3407Biophysical Journal Volume 77 December 1999 3407–3423

of the acrosomal process is only one step (Colwin andColwin, 1955; Dan, 1967; Inoue and Tilney, 1982; Tilneyand Inoue, 1982). Fig. 1, adapted from Figs. 1 and 6 ofInoue and Tilney (1982), shows a schematic of the threestages in the reaction. In the untriggered sperm (Fig. 1 a),actin is located in the periacrosomal region (P), behind theacrosomal vacuole (V). The actin is bound in an insolublecomplex with profilin and two high molecular weight pro-teins (Tilney, 1979; Tilney and Inoue, 1985).

In nature, the acrosomal reaction is triggered by contactwith Thyone egg jelly, but it can also be triggered in thelaboratory with ionophores (Inoue and Tilney, 1982). Sev-eral events take place when the reaction is triggered. Move-ment of Ca2� into the sperm induces the fusion of the frontmembrane of the acrosomal vacuole (V) and the frontmembrane of the sperm (Tilney, 1979; Fig. 1, a and b). The

fused membrane subsequently opens, releasing the contentsof the vacuole (Inoue and Tilney, 1982). At this point, therear membrane of the vacuole becomes the front membraneof the acrosomal process.

Then something striking takes place: the periacrosomalregion doubles in volume (Inoue and Tilney, 1982). Thedoubling occurs rapidly, in 50–70 ms, indicating not onlythat there is a strong driving force for water to enter theperiacrosomal region, but also that the surrounding mem-brane is quite permeable to water. The water movement ispresumably driven by an osmotic pressure change. Al-though actin is released into solution in the periacrosomalregion after the acrosomal reaction is triggered (Tilney etal., 1978), its concentration is not high enough to causedoubling of the compartment’s volume, even taking intoaccount the effects of nonideality. Salt, however, is presentin much higher concentrations. In another echinoderm, thesea urchin, the acrosomal reaction is accompanied by a 30mV membrane depolarization that is caused by the influx ofions into the acrosome (Schackmann et al., 1981). Ions alsoflood the acrosome of Thyone, and although the membranepotential has not been measured, it is probably this influxthat leads to the observed water influx and volume doubling.

Once actin unbinds from the high molecular weight pro-teins, polymerization begins. The profilin-actin complexpolymerizes onto the actomere (A in Fig. 1), a nucleatingorganelle (Tilney, 1978), and the profilin is released intosolution. The extension of the process then proceeds rapidly(Fig. 1 c). Decoration with myosin subfragments revealsthat these actin filaments are uniformly oriented with theirbarbed (fast-growing) ends away from the actomere (Tilneyand Kallenbach, 1979). The profilin-actin complex presum-ably diffuses to the tip of the process where the actin isadded to the process. Because nucleotide exchange occursfar more rapidly on profilin-actin than on actin alone (The-riot and Mitchison, 1993), the monomers are expected to bepolymerization-competent on arrival at the tip. Processestriggered by ionophores in the laboratory have reachedlengths of 30–90 �m in only 5–10 s (Tilney and Inoue,1982), with growth ceasing rather suddenly.

Several groups have calculated the extension speed of theacrosomal process assuming that actin is delivered by dif-fusion alone. A model of Tilney and Kallenbach (1979),based on the work of Hermans (1947) and corrected byPerelson and Coutsias (1986), assumes that diffusion deliv-ers actin monomers to the growing filament tips whereinfinitely fast polymerization takes place. A second modeldue to Perelson and Coutsias (1986) assumes a finite poly-merization rate and includes effects due to fluid movementinto the process. Because both models predicted growthrates significantly less than those observed experimentally,it was then believed that diffusion alone could not deliveractin rapidly enough to explain the growth of the process.

The diffusion models noted above are oversimplified.First, neither model takes into account that actin filamentsoccupy space that could otherwise be used to transport actin.Including the volume of the actin filaments will reduce the

FIGURE 1 A schematic of Thyone sperm and its acrosomal process(based on Fig. 1 and Fig. 6 of Inoue and Tilney, 1982). Panel (a) shows theintact sperm. The periacrosomal region (P) holds the unpolymerized actinin an inert form between the acrosomal vacuole (V) and the nuclearmaterial (N). The mitochondria (M) are located near the tail of the sperm,which is only partially pictured. In panel (b), the acrosomal reaction hasbeen triggered, and the acrosomal process has started to grow. The frontmembrane of the acrosomal vacuole (V) and the front membrane of thesperm (Tilney, 1979) have fused, and the contents of the vacuole (Inoueand Tilney, 1982) have been released. The rear membrane of the vacuolehas become the front membrane of the acrosomal process. Actin polymer-ization has been nucleated on the actomere (A). The filaments are orientedwith their barbed ends pointing away from the body of the sperm. Mono-mers diffuse from the periacrosomal region to the tip of the process, wherethey are added to the filaments. In (c), the acrosomal process has reachedits final length, between 30 and 90 �m. The process, shown with greatlyexaggerated thickness, is �50 nm in diameter and contains �60 filamentsin cross-section. The final shape and size of the periacrosomal region arenot known.

3408 Biophysical Journal Volume 77 December 1999

diffusive flux and the extension speed. Second, both modelsignore the volume change in the acrosome at the onset of theacrosomal reaction. Because the periacrosomal region dou-bles in volume before the process begins to extend, theinitial actin concentration immediately drops by half, dra-matically reducing the diffusive flux to the tip of the pro-cess. Third, both models ignore the depletion of actin in thereservoir as the process grows. This depletion will furtherreduce the diffusive flux of actin.

In the next section we introduce a diffusion-only modelfor the extension of the acrosomal process that correctsthese flaws of previous models. Of course, this correcteddiffusion-only model reproduces the experimental data evenless well than the earlier models. In the Actin Reconcentra-tion Model we extend the corrected model to include activeion transport, which leads to higher actin concentration,increased diffusive flux, and faster extension speed. In theActin Reconcentration Model, we also take account of theextreme nonideality due to high protein concentrations inthe cell. Using parameter values from the literature, thismodel predicts behavior within the range of the experimen-tal data.

In the fourth section, we present more detailed versions ofthe models in the second and third sections. Many of theapproximations made in the initial calculations are relaxed,and the resulting equations are solved iteratively. The re-sults of the more detailed models are shown to be in excel-lent agreement with the initial calculations, thereby justify-ing the approximations made earlier. In the final sections,we discuss implications and interpretations of the models,and we summarize our conclusions.

DIFFUSION-ONLY MODELS

Minimal models

The system consists of a cylindrical process of radius r andthe periacrosomal region, which acts as a “reservoir” ofactin and solvent for the growing process. The reservoir’sinitial volume is known from experimental observations.Actin monomers diffuse from the reservoir to the tip of theprocess, where they polymerize onto the ends of the fila-ments. If the concentration of monomers in the reservoir isassumed to be constant, and chemical steps involved inpolymerization are infinitely fast, the growth of the acroso-mal process can be mapped onto a previously solved diffu-sion problem (Hermans, 1947). The length of the acrosomalprocess as a function of time is then given by Tilney andKallenbach (1979)

L�t� � z�4Dat, (1)

where Da is the diffusion coefficient of the actin monomer.The constant z is given by the solution to

z exp�z2�erf�z� �ca,res

��

�

n , (2)

where ca,res is the concentration of actin monomers in thereservoir, � is the average increase in filament length permonomer added, n is the number of growing filaments perunit cross-sectional area, and erf(x) is the error function(Arfken, 1985). The concentration of actin monomers in theprocess, at a distance x from the reservoir, is given by

ca�x, t� � ca,res�1 �erf�x/�4Dat�erf�L/�4Dat�

� (3)

for x � L(t).Perelson and Coutsias (1986) later improved this model

by relaxing the assumption of infinitely fast polymerizationand including the effects of the movement of fluid into theextending process. Their more complicated singular pertur-bation analysis gave the same L � �t form as before, buttheir numerical prefactor was smaller.

The forgoing models are difficult to extend because anyimprovements would probably render them analytically un-solvable. We will formulate a model that is simpler math-ematically but gives similar results. This model will then bemodified in later sections to include some of the features ofthe physical system that the previous models ignored.

Inasmuch as it is clear that actin delivery, not actinpolymerization, is the limiting factor for growth in thissystem, we continue to assume that polymerization is infi-nitely fast. Any actin monomer that diffuses to the end ofthe process is assumed to be immediately polymerized. Theflux of monomers added to the filaments is thus set equal tothe diffusive flux of monomers to the tip: jpoly � jdiff(L).

The diffusive flux is given by Fick’s law: jdiff � Da�ca/�x. In this system, the actin concentration profile ca(x)equilibrates quickly. We therefore assume that the actinmonomer concentration always reaches the linear profilebetween the reservoir and tip that it would have in a steady-state, constant length system. The concentration of actin atthe tip ca,tip is set to zero due to the assumption of infinitelyfast polymerization, so the concentration profile is ca(x) �ca,res(1 x/L). That this profile is reasonable can be seen byexamining Eq. 3. The error function erf(x) is fairly linear forx � 1. Replacing erf(x) by a linear function in Eq. 3 givesa linear concentration profile. The diffusive flux is then

jdiff � Da

�ca

�x �Daca,res

L . (4)

The flux of monomers jpoly being polymerized is directlyrelated to the extension speed �ext by

�ext � ��/n�jpoly. (5)

Since �ext � dL/dt and jdiff � jpoly, Eq. 5 becomes afirst-order differential equation:

dLdt � ��

n�jdiff � ��

n�Da

L ca,res. (6)

Olbris and Herzfeld Actin Delivery in the Acrosomal Process 3409

With the initial condition L(0) � L0, Eq. 6 can be integratedto

L�t� � �2 Daca,res�

n t L02. (7)

Equation 7 is much simpler in form and derivation than Eqs.1 and 2. However, as will be shown in our results, thepredictions are nearly identical.

Corrected model

From this starting point we may now correct some flaws inthe minimal models. One significant omission in the mini-mal models is that the volume occupied by the actin fila-ments is ignored. Actin filaments are densely packed in theacrosomal process, and the available volume through whichmonomers diffuse to reach the filament tips is reduced. Wereplace the number of growing filament ends per area n withthe number of filament ends per fluid-accessible area Nf/Ad,where Nf is the total number of filaments in the process(Nf � n�r2), and Ad is the cross-sectional area of the processthat is available for diffusion (i.e., not occupied by filaments).

The other changes are in the treatment of the reservoir.First, the reservoir volume swells to approximately twice itsinitial volume: V03 2V0. Second, the concentration of actinmonomers in the reservoir is not constant. Initially,ca,res(0) � Na0/2V0, where Na0 is the initial number of actinmonomers in the reservoir. The number of monomers in thereservoir then drops by the number of monomers that haveleft. If the relatively few free monomers in the process areignored, the actin concentration in the reservoir is decreasedby the number of monomers that have been added to theactin filaments:

ca,res�t� �Na0 � Nf�L/��

2V0� ca,res�0��1 � L/L�, (8)

where L � Na0�/Nf is the length the process would attainwhen all the monomers in the reservoir are polymerized.

When all of the above corrections are collected, Eq. 6becomes

dLdt � ��Ad

Nf�Da

LNa0 � Nf�L/��

2V0�DaAd

2V0

�1 � L/L�

L/L. (9)

This differential equation can be integrated with the initialcondition L(0) � L0 to give

L0 � LL

� ln� 1 � L/L

1 � L0/L� � � DaAd

2V0L�t , (10a)

which cannot be solved for L(t) explicitly. Because L � L,we may use the expansion ln(1 x) � �m�1

xm/m, to findthe following expression for t(L):

t�L� � �2V0L

DaAd� �m�2

1m�� LL

�m � �L0

L�m� . (10b)

If only the first term of the series is retained, the t� L2 formis recovered (compare Eq. 7). In any case, Eq. 10a can besolved numerically for L(t) by software packages such asMathematica (Wolfram, 1996).

Choice of parameter values

Table 1 summarizes the values of the various parametersused in calculating extension speeds under the diffusion-only models. In reality, the diffusing solute is not actin, butthe actin-profilin complex. Its diffusion coefficient Da is notknown. Estimates for the diffusion coefficient of actin varyfrom 5 to 8 � 107 cm2/s in aqueous solution (Lanni et al.,1981; Lanni and Ware, 1984; Tait and Frieden, 1982).Noting that human hemoglobin, whose molecular weight is�10% greater than that of profilin and actin together, has adiffusion coefficient of �5 � 107 cm2/s in aqueous solu-tion (Gros, 1978; Muramatsu and Minton, 1988), we takeDa � 5 � 107 cm2/s.

Actin filaments are double-stranded, so the average ex-tension per monomer added is about half the monomerlength: � � 2.7 nm (DeRosier and Tilney, 1984). Electronmicrographs show �Nf � 60 filaments in the cross sectionof a Thyone process with r � 0.025 �m (Tilney and Inoue,1982), so the density of growing ends is n� Nf/�r2 � 3.1 �1012/cm2. Some of the cross-sectional area of the process isoccupied by the polymerized actin and is not accessible tofluid. Taking the density of protein to be 1.4 g/cm3 (Cantorand Schimmel, 1980), an actin monomer of 42 kDa then hasa volume of �5.0 � 1020 cm3. Therefore, the Nf � 60polymerized monomers in a � � 2.7 nm length of theprocess occupy �3.0 � 1018 cm3, or �1.1 � 1011 cm2

of cross-sectional area, leaving Ad � 8.5 � 1012 cm2 of thetotal �r2 � 2.0 � 1011 cm2 accessible for diffusion. Thiscorresponds to an average diameter of the actin doublestrands that is smaller than the maximum diameter reportedfrom atomic models because we are concerned with thevolume occupied by the filament rather than the distancebetween adjacent filaments.

The periacrosomal region, which holds the actin beforethe process starts to elongate, is estimated to have an initial,pre-doubling volume of V0 � 5.1 � 1013 cm3 (Tilney andInoue, 1982). Within this volume, actin (42 kDa) is stored in

TABLE 1 Parameter values for the diffusion-only models

Symbol Parameter Value

Da Actin diffusion coefficient 5.0 � 107 cm2/s� Length increment per actin monomer

polymerized2.7 nm

Nf Number of actin filaments 60r Thyone process radius 0.025 �mAd Cross-sectional area accessible for

diffusion8.5 � 1012 cm2

V0 Initial reservoir volume 5.1 � 1013 cm3

Na0 Initial number of actin monomers inthe reservoir

3.1 � 106

L0 Initial length of the process 0.19 �m


an insoluble form, along with profilin (16 kDa) and twohigher molecular weight proteins (220 and 250 kDa) in a1:1:1/12:1/12 ratio (Tilney and Inoue, 1982). If 70% of V0

(midway between face-centered and body-centered closepacking) is occupied by those proteins in the given ratio,with an average density of 1.4 g/cm3, then the reservoirinitially contains �Na0 � 3.1 � 106 monomers of actin.That is in excess of the 2 � 106 monomers required to builda 60-filament, 90-�m-long process. After the volume of theperiacrosomal region doubles at the onset of the acrosomalreaction, the actin concentration in the reservoir will beNa0/2V0, or �ca,res(0) � 5 mM. If the swelling is ignored,the concentration is twice as large, 10 mM.

The initial length of the process L0 is set to 0.19 �m. Thatparticular number is unimportant for this model, but it ischosen for convenience in later comparisons with othermodels.

Results

We now compare the predictions of the above models toexperimental data. Unless otherwise noted, the results forthe models were calculated using the parameters in Table 1.

Fig. 2 shows the length of the acrosomal process (�m)plotted as a function of time (s) for the experimental dataand the models. The experimental data are taken from Fig.2 of Tilney and Inoue (1982). Curve (b) is the prediction ofthe earlier model (Eqs. 1 and 2) (Hermans, 1947; Tilney andKallenbach, 1979) that ignores the volume of the actinfilaments, the activity of actin, and the swelling of thereservoir [L(t) � L0 � z�4Dat is plotted with z as in Eq. 2.L0 was added to Eq. 1 to allow comparison to the simpler

models, which are singular for L0 � 0.] Our simpler diffu-sion-only model (Eq. 7), with the same flawed features,gives curve (a). The two models have nearly identical pre-dictions for these parameters, justifying the simplifyingassumptions that we have made.

Fig. 3 shows the experimental data, curve (a) from Fig. 2,and curves (c)–(f), which show the effects of including thevarious corrections to the minimal models. Curve (c) showsthe effect of including actin monomer depletion in thereservoir. The effect is small when the process is not long.For curve (d), the reservoir has been allowed to swell totwice its initial volume, halving the initial concentration ofactin, and in curve (e), the fluid-accessible volume of theprocess is reduced to account for the presence of the actinfilaments. Each of the curves (c)–(e) shows the effect of onecorrection; the combined effect of all three corrections isshown by curve (f). All of the predictions of these modelsfall below the experimental data.

THE ACTIN RECONCENTRATION MODEL

Effects of water movement

The doubling in volume of the periacrosomal region at theonset of the acrosomal reaction suggests that water move-ment is an important feature of this system. The ActinReconcentration Model quantifies how water movement canaffect actin delivery and process extension speed.

At the beginning of the acrosomal reaction, ion channelsopen and allow salt into the periacrosomal region. Waterfollows, and the periacrosomal region doubles in volume.We assume that the channels eventually return to their

FIGURE 2 Plots of length of the acrosomal process (�m) versus time (s)for the diffusion-only models using parameters from Table 1 and experi-mental data from Fig. 2 of Tilney and Inoue (1982). We keep the originallettering for the experimental data: B (E), J ({), D (�), C (‚), F (�), andA (�). Curve (a) is the result of Eq. 7, and curve (b) is the result of anearlier model based on Hermans’ equation (Eq. 1) (Tilney and Kallenbach,1979).

FIGURE 3 Plots of length of the acrosomal process (�m) versus time (s)for theory and experiment. The data and curve (a) are as in Fig. 2. Curves(c)–(e) show the effect of various corrections: depletion of actin monomersfrom the reservoir (c), initial swelling of the reservoir (d), volume occu-pation by actin filaments in the process (e). The combined effect of all threecorrections is shown in curve (f).


initial states, and that the normal salt content of the com-partment is gradually reestablished by ion pumps.

In the model, we artificially divide the salts in the systeminto two types: -salt, whose cations are initially diluteinside the cell, flood in through opened ion channels, andare later pumped out again; and �-salt, whose cations arepresent in the cell initially and do not enter or leave duringthe acrosomal reaction. In Thyone, the -cation is thought tocorrespond to Na� (Tilney and Inoue, 1985), and the �-cat-ion is thought to correspond to K�. The most likely candi-date for the ion pump is the Na-K ATPase. This ubiquitouspump exports three Na� ions and imports two K� ions foreach molecule of ATP hydrolyzed, while electroneutrality ispreserved by passive anion movement.

As the pumps work to restore the initial salt composition,water leaves the system through the membrane to maintainosmotic equilibrium. Actin in the reservoir becomes moreconcentrated, and diffusion becomes more rapid. See Fig. 4for a schematic of the Actin Reconcentration Model.

The simple diffusion-only model in the second sectionneeds only to be adjusted slightly so that the volume of thereservoir changes as ions are pumped out. At t � 0, it isassumed that N0 -salt ions have already entered the peria-crosomal region, and water moving to maintain osmoticequilibrium across the membrane has already swollen theregion to Vres(0) � 2V0. These processes are not explicitlymodeled. As the ions are pumped out, N(t) decreases, andwater leaves the system to maintain osmotic equilibrium.Because the original doubling of the reservoir occurred inonly �60 ms (i.e., a volume V0 of fluid enters in 60 ms)

(Tilney and Inoue, 1982), we assume that water can move toequalize osmotic pressure differences infinitely fast as saltis pumped out of the system. Because the process is so thin,we ignore its volume and any solutes within its volume.Therefore the volume of the reservoir is determined by thetotal number of salt ions within the reservoir:

Vres�t� � V0�N�0 N�t�N�0

� , (11)

where N(t) is the number of excess salt ions remaining inthe reservoir, and N�0 is the unvarying number of �-salt ionsin the reservoir. The number of -salt ions in the reservoiris governed by

dN

dt � S0j, (12)

where S0 is the area of the membrane through which ionsare being pumped, and j is the flux (in number of ionspumped per time per area) with which salt is pumpedthrough the membrane. Although the reservoir’s initial vol-ume is known from experimental work, its initial membranearea must be estimated. In the absence of any experimentaldata about membrane creation or distribution, the system’stotal membrane area (reservoir plus process) is assumedconstant.

As a simple approximation to the workings of an ionpump, we take the flux of ions j through the membrane tobe

j � j-max� c

c c,1/2� , (13)

where c is the concentration in the reservoir of the ionsbeing pumped, j-max is the maximum capacity of thepumps, and c,1/2 is the concentration at which the ionpumps operate at half of their maximum capacity. Theconcentration of ions is c � N/Vres, where the volume ofthe reservoir is now a function of time. Equation 12 thenbecomes

dN

dt � S0j-max

N

N c,1/2Vres. (14)

With N(0) � N0, Eqs. 11 and 14 can be solved numeri-cally for Vres.

The only adjustment that then needs to be made to Eq. 9is to replace (2V0) by Vres:

dLdt �

DaAd

Vres

�1 � L/L�

L/L. (15)

This equation cannot be integrated, but it can again besolved numerically by Mathematica (Wolfram, 1996).

Effects of nonideality

Our second modification in the Actin ReconcentrationModel is to the driving force for diffusion. Because the

FIGURE 4 Schematic of the Actin Reconcentration Model. (a) Actin insolution in the periacrosomal region diffuses to the tip of the process whereit is polymerized. When salt ions are pumped out, water follows, causingthe reservoir to shrink (b). Actin becomes more concentrated, increasingdiffusive flux. The process and monomers are shown exaggerated in sizecompared to the reservoir.


volume fraction of protein is very high in this system, thenonideality of actin is significant. We replace the concen-tration of actin with its activity:

ca 3 aa � �aca, (16)

where aa is the activity of actin and �a is the activitycoefficient of actin. The activity coefficient is derived fromthe chemical potential �a via �a � aa/aa,id � exp(�a �a,id/kBT), where aa,id and �a,id are the ideal activity andchemical potential, respectively. According to scaled parti-cle theory (Han and Herzfeld, 1994), the chemical potentialof a monodispersed hard sphere fluid is

�a

kBT� ln�ca a

3� � ln�1 � a� 7� a

1 � a�

�152 � a

1 � a�2

3� a

1 � a�3

,

(17)

where a is the thermal wavelength of actin, a � ca�a is thevolume fraction of actin, and �a is the volume of an actinmonomer. The ideal value is obtained in the limit a 3 0,and the activity coefficient is

�a� a� �1

�1 � a�exp�7� a

1 � a�

152 � a

1 � a�2

3� a

1 � a�3� .

(18)

With the replacement of the actin concentration ca �(Na0/Vres)(1 L/L) by the activity aa � ca�a( a) �ca�a(ca�a), Eq. 15 becomes

dLdt �

DaAd

Vres

�1 � L/L�

L/L�a�Na0�1 � L/L�

Vres�a� . (19)

As before, this equation can be solved numerically byMathematica (Wolfram, 1996).


Table 2 summarizes the values of the parameters used in theActin Reconcentration Model that are not included in Table1. The number of ions pumped across the membrane de-

pends on its surface area. At least 1.4 � 107 cm2 ofmembrane must be present to cover a 0.05-�m diameter,90-�m-long cylindrical process. This is more than is re-quired to encompass the initial reservoir after swelling (butbefore the process is constructed). The final volume of theperiacrosomal region is not experimentally known, but inthe Actin Reconcentration Model, it reaches a final volumeclose to its pre-doubling volume V0. About another 3.1 �108 cm2 of membrane would be needed to cover a sphereof V0 � 5.1 � 1013 cm3. We use an initial membrane area15% larger than the sum of the membrane needed to coverthe process and reservoir separately: S0 � 2 � 107 cm2.

We estimate the rate of ion pumping from experimentaldata. The maximum reported turnover for Na-K ATPases is�5 � 1013 ATP/(cm2 � s) in rat soleus muscle (Sejersted,1988). Assuming net movement of three Na� ions and threeaccompanying anions per ATP hydrolyzed (while higherK� permeability yields no net K� transport), this turnovercorresponds to a net flux of ions moving across the mem-brane of j-max � 3 � 1014 ions/(cm2 � s). The maximum ionflux could be higher, because neither the density of thepumps in this membrane nor their maximum turnover rate isknown. The other parameter controlling ion transport, theconcentration at half rate c,1/2, is set to 17 mM. The valueof c,1/2 affects the rate at which the ion transport falls withdecreasing salt concentration, but the extension rate of theprocess is insensitive to the value of c,1/2 as long as it issufficiently small.

The external osmotic pressure �ext/kBT is set to 1 M,approximately the osmotic pressure of seawater (Lide,1998), the natural environment of a sea cucumber. Thenumber of ions initially present in the reservoir (�-salt ions)can be calculated by assuming that the periacrosomal regionis in osmotic equilibrium prior to the acrosomal reaction.Given the (pre-doubling) volume, we calculate that theremust be N�0 � 3.06 � 108 ions present to balance theexternal osmotic pressure. When the acrosomal reaction istriggered, -cations enter, and anions follow. Assuming thatsomething close to a state of osmotic equilibrium is attainedafter the doubling, an influx of N0 � 3.06 � 108 ionswould be required to balance the external osmotic pressurein the added volume (N0/V0 � �ext/kBT).

The activity coefficient of the profilin-actin complex iscalculated using Eq. 18, which requires the complex’s mo-lecular volume va. Again assuming a density of 1.4 g/cm3,the profilin (16 kDa)-actin (42 kDa) complex would have avolume of �va � 6.9 � 1020 cm3 � (4.1 nm)3.

Results

As in the second section, the theoretical results are com-pared with the experimental data of Tilney and Inoue(1982). Unless otherwise noted, the parameters in Tables 1and 2 were used in calculating the results of the ActinReconcentration Model.

Fig. 5 shows the effect of including pumping and non-ideality in the model. Fig. 5 b is identical to Fig. 5 a except

TABLE 2 Additional parameter values for the ActinReconcentration Model


S0 Initial reservoir surface area 2.0 � 107 cm2

j-max Maximum ion flux 3.0 � 1014/(cm2 � s)c,1/2 Ion concentration at half-maximum

flux17 mM

�ext/kBT External osmotic pressure 1 MN0 Number of ions entering during

swelling3.06 � 108

N�0 Initial number of ions in the reservoir 3.06 � 108

�a Volume of profilin-actin complex 6.9 � 1020 cm3


that b is plotted on a linear-linear scale, and the unprimedcurves are omitted. The first (unprimed) set of curves in Fig.5 a shows the effects of pumping while preserving ideality(solution of Eq. 15). Curve (f) is the same as in Fig. 3,showing the corrected diffusion-only model (i.e., j-max �0). Curve (h) was calculated with j-max � 1026 ions/(cm2 �s). That value is chosen to be sufficiently large that thepumps are effectively infinitely fast. Because the initialswelling of the reservoir is counteracted immediately, thiscase is equivalent to a “no swelling” scenario. The extensionrate is increased due to the increased concentration of actinmonomers in the unswollen reservoir. Curve (g) was calcu-lated with the physiological j-max � 3 � 1014 ions/(cm2 �s). It falls neatly between curves (f) (no pumping) and (h)(infinitely fast pumping), reflecting that as ions are pumpedout, the reservoir volume decreases from 2 V0 to �V0. Thepredictions, however, still fall well below the experimentaldata.

The effects of nonideality are added to the model in theprimed curves in Fig. 5, a and b. These curves represent thesolution to Eq. 19 with the same parameters as for thecorresponding unprimed curves. The sole difference is theinclusion of nonideality in the primed curves. The brokenline is the same as curve (g�), but with a slightly higherpump rate: j-max � 5 � 1014 ions/(cm2 � s).

The data and the models each have a characteristic L(t)shape. The experimental data have been characterized asfalling on a straight line when plotted as L2 vs. t (Tilney andInoue, 1982). On a log-log plot (Fig. 5 a), such a curvewould be straight. However, it is clear in Fig. 5 a that theexperimental data generally have a concave-down shape.Furthermore, in some of the trials, sigmoidicity can be seenat shorter times. The linear-linear plot, Fig. 5 b, provides amore realistic representation for short times, where mea-surements of shorter processes entail larger relative errors.

Some of the calculated curves ((f), (f�), (h), and (h�)) inFig. 5 a appear to be straight. This is an artifact of plottingthe curves only over experimentally relevant times (0.5–10s). In fact, all of these curves have a downward concavity atlarge t, due to depletion of monomers in the reservoir. Thiscan be seen when they are plotted over a wider time interval(Fig. 5 c). In the models corresponding to these curves, thereservoir volume is constant, either due to no ion pumping(curves (f) and (f�)) or infinitely fast ion pumping (curves(h) and (h�)). When the reservoir volume is constant andnonideality is not included (curves f and h), the governingdifferential equation is Eq. 9, whose solution is Eq. 10(although curve (h) was generated via Eq. 15 with an“infinite” pump rate). When the process length L is small,t(L) in Eq. 10b is well-approximated by the first term (m �2) in the series, and the L2 � t form of Eq. 7 is recovered.

FIGURE 5 Plots of length of the acrosomal process (�m) versus time (s)for theory and experiment on (a) log-log and (b) linear-linear scales, and(c) an expanded log-log scale. The data are as in Figs. 2 and 3, and curve(f) is as in Fig. 3. Curves (g) and (h) show results for the Actin Recon-centration Model ignoring nonideality (the solution to Eq. 15). Curve (h)was calculated assuming effectively infinitely fast ion pumping. The infi-nitely fast pumping negates the effects of the initial swelling. Curve (g) wascalculated with j-max � 3 � 1014 ions/(cm2 � s), i.e. the unprimed curvesare omitted in b. Curves (f�)–(h�) were calculated including the effects ofsolute nonideality (the solution to Eq. 19), with parameters that are other-wise the same as for the unprimed curves. The dashed curve is the same ascurve (g�), except the pump rate has been increased to j-max � 5 � 1014

ions/(cm2 � s). Panel (c) was calculated with a smaller L0 than the otherpanels so as to better resolve the behavior at small t. The large t behavioris unaffected by using a smaller L0. The dashed box in (c) is the plot areashown in panel (a).


So in Fig. 5 c, (f) and (h) are essentially straight with slope2 at small t, and it is only at larger t that the downwardconcavity can be seen. Even then, the curvature is small.However, when nonideality is included, as in curves (f�) and(h�), the same approximation does not hold, and the down-ward concavity occurs most visibly at shorter times thanthose plotted in Fig. 5, a and b.

The effects of varying the parameters in the model canalso clearly be seen in Eq. 10b. Increasing either the diffu-sion coefficient Da or the area available for diffusion Ad

results in a shorter time to reach the same length (fastergrowth). Increasing the reservoir volume V0 has the oppo-site effect, because a larger reservoir volume translates to alower concentration of actin, and therefore a shallowerconcentration gradient to the tip. The parameter L, which isproportional to the number of actin monomers Na0 initiallypresent in the reservoir, occurs strictly in the denominatorbecause the series starts with m � 2. Increasing L corre-sponds to increasing the actin concentration and thereforedecreases the time needed to reach a given length.

The curves in Fig. 5 a corresponding to the Actin Recon-centration Model, (g), (g�), and the broken line, have asigmoid shape. The position of the downward concavity inthe calculated curves, which appears at longer times as inthe data, is controlled in large part by the product of the ionpump rate j-max and the system’s membrane surface areaS0. Much of the experimental data can be bracketed bycurve (g�) and the broken curve, corresponding to j-max �3 and 5 � 1014 ions/(cm2 � s) respectively, with all of theother parameters held constant. The point in the ActinReconcentration Model where the slope begins to flattencorresponds roughly to the point where the reservoir stopschanging volume because salt concentrations have beenrestored.

Most of the other parameters in the model affect theoverall magnitude and position of the L(t) curve more thanits shape. Note that dL/dt in Eq. 15 is the growth rate.Therefore, increasing either Da or Ad will increase thegrowth rate, just as discussed above with respect to Eq. 10.Increasing L has the same effect. Any of these increases inthe growth rate tend to move the L(t) curve upward. Also, asmentioned above, the product j-maxS0 controls the positionof the inflection in the L(t) curve, but an increase in thatcombination also increases the growth rate and moves theL(t) curve up, because a larger j-maxS0 results in a Vres thatdecreases faster.

The behavior of variables other than L(t) better illustratesthe relationship between ion pumping and process growth.Fig. 6 a shows plots of the number of -salt ions (N), thevolume of the reservoir (Vres), the concentration of actinmonomers (ca), and the process length (L) as a function oftime in the Actin Reconcentration Model, using the param-eters for curve (g�) in Fig. 5. The salt in the reservoir dropssteadily due to the action of the ion pumps until �t � 5.5s, at which time nearly all of the excess ions have beenpumped out of the system. The volume of the reservoirshrinks in direct correlation to the number of -salt ions.

The concentration of actin monomers in the reservoir risesas the volume shrinks. Therefore, the driving force for actindelivery also increases. The effects can be seen in theupward bend in L(t) at �t � 3–5 s. At �t � 5.5 s, Vres(t)

FIGURE 6 Reservoir behavior in the Actin Reconcentration Model: thevolume of the reservoir (Vres), scaled by its pre-swelling value; the con-centration of actin monomers in the reservoir (ca), scaled by its initialvalue; the concentration of excess ions in the reservoir (c), scaled by itsinitial value; and the length of the acrosomal process (L), scaled by its finalvalue. The curves in (a) were calculated in the Actin ReconcentrationModel using the same parameters as were used to generate Fig. 5, curve(g�). In panel (b), neither ions nor water are transported across the mem-brane. The parameters are the same as were used to generate Fig. 5, curve(f).


plateaus because no more excess salt is left to be pumpedout. The concentration of actin monomers no longer risesdue to the shrinkage of the reservoir. Rather, it begins tofall, as monomer depletion due to polymerization is nolonger masked by the reconcentration of actin due to reser-voir shrinkage. At this point, L(t) turns downward again.When ion transport is turned off (diffusion-only model),most of these effects are absent. Fig. 6 b shows the samecurves as Fig. 6 a when ion pumping has been turned off.The reservoir’s volume remains constant, so the actin con-centration in the reservoir drops due to polymerizationrather than rising due to reservoir shrinkage.

RELAXATION OF ASSUMPTIONS

Method

Numerical methods allow us to relax some of the assump-tions made above and assess their validity. The mathemat-ical base of this approach is the continuity equation. Be-cause the process is very narrow (�50 nm in diameter)compared to its length (30–90 �m) (Tilney and Inoue,1982), lateral diffusion quickly levels any lateral concen-tration gradients. Therefore, the concentration of actinmonomers and salt ions can be considered constant acrossthe process diameter, and gradients occur only along theprocess. Fluid flow in the process is also assumed to beone-dimensional. Instead of solving the full low Reynoldsnumber hydrodynamic equations, the fluid speed is assumedto be uniform over the cross-sectional area of the process,although varying along its length. The one-dimensionalcontinuity equation for the solutes is then

�ci�x, t��t

�ji�x, t��x � 0, (20)

where ci(x, t) are the solute concentrations, ji(x, t) are thesolute fluxes (in particles per area per time), and i � a, �,and refers to actin and the two varieties of salt, respec-tively. The fluxes arise from diffusion and bulk flow:

ji � Di��ai/�x� �ci, (21)

where Di is the diffusion coefficient of the ith species, � isthe speed of the movement of fluid in the process, ai � �ici

is the activity of the ith species, and �i is the activitycoefficient of the ith species. In practice, we make theapproximation that the salt ions are ideal solutes: a� � c�

and a � c.We must also add sink terms i to the continuity equation

to account for particles that leave the solution throughpolymerization (actin) or active transport (-salt).

�ci

�t �ji�x� i (22)

If we assume that the fluid speed varies slowly along theprocess, the continuity equation then takes the form

�ci

�t ��ci

�x � Di

�2ai

�x2 � i, (23)

Because this equation is an expression of the conservationof mass, it holds everywhere in the process. The �-salt ionsdo not enter or leave the process, so � � 0 everywhere.

accounts for the loss of ions through the action of ionpumps. It can be easily calculated using Eq. 13 (see theAppendix for details).

For actin, a � 0 only at the tip of the process wherepolymerization takes place. a is proportional to the poly-merization rate, which is proportional to the extension speeddL/dt. Unlike the earlier models, in which actin was as-sumed to polymerize infinitely fast, in this model the rate ofactin polymerization is given by the Brownian RatchetModel (Peskin et al., 1993) of force transduction (see Fig.7). In the Brownian Ratchet Model, polymerization “ratch-ets” membrane fluctuations into unidirectional motion.When the ends of actin filaments abut the cell membrane,polymerization is blocked. However, fluctuations of themembrane may open gaps between the filament and themembrane, allowing polymerization to occur. Then themembrane cannot return to its original position due to thepresence of the now-longer filament. In the range of param-eters relevant to this system, the Brownian Ratchet Modelpredicts an extension speed dL/dt of

dL/dt� ��k�aa,tipe� � k�, (24)

where k�(k) is the on (off) rate constant for actin poly-merization, and � � f�/kBT, where kB is Boltzmann’s con-stant, T is the absolute temperature, and f is the force oneach filament opposing elongation. The Brownian Ratchet

FIGURE 7 A schematic of the Brownian Ratchet Model (Peskin, et al.,1993). (a) An abutting membrane hinders polymerization, but (b) Brown-ian fluctuations can create membrane-filament gaps large enough for amonomer to be added; after which (c) the membrane cannot return to itsoriginal position: the “ratchet.”


is not a critical part of our model. We chose the BrownianRatchet Model as a convenient way to quantitatively relateextension speed to local monomer concentration and oppos-ing force, but any quantitative model for protrusion wouldsuffice. The details are unimportant, because extensionspeed is limited by actin delivery, not polymerization rate ormembrane tension. However, including a finite polymeriza-tion rate allows us to test the initial assumption of infinitelyfast polymerization.

The analytical model presented in the previous sectiondid not include one effect of allowing water to movethrough the process membrane: the effects of bulk fluidflow within the process. The speed � of the fluid flow in Eq.23 is not in general equal to the extension speed of theprocess. It is also determined by how much water leaves theprocess through the membrane. Because the actin filamentsthat form the core of the acrosomal process prevent theprocess’s membrane from collapsing, escaping water mustbe replaced from elsewhere in the system: ultimately fromthe periacrosomal region, which has no actin filamentframework and is free to shrink as fluid leaves it. At anygiven location in the process, the fluid speed is determinedby the volume of fluid that passes that point on its way fromthe periacrosomal region to the rest of the process, either tofill the growing process or to replace water that has left theprocess through the membrane. This volume is divided bythe cross-sectional area of the process and the time intervalto give the fluid speed at that point in the process. This bulkflow potentially transports more actin to the tip of theprocess, but because the membrane area of the reservoir isfar larger than that of the process (at least during the earlystages of growth), relatively little water leaves the process,and little actin is transported in this manner. By the time theprocess’s membrane area becomes appreciable, -saltpumping is finished, and no more water leaves the process.Then the fluid speed in the process is the same as theprocess extension speed.

In general, the flux jw of water molecules through amembrane due to an osmotic pressure difference is given byKatchalsky and Curran (1967) and Macey and Brahm(1989)

jw � Pw��, (25)

where Pw is the hydraulic permeability coefficient of themembrane. �� is the osmotic pressure difference across themembrane:

�� ext � c� � c, (26)

where �ext is the osmotic pressure of the medium outsidethe process membrane. Even taking the nonideality of actininto account, the two cation types and their accompanyinganions can be assumed to contribute almost all of theosmotic pressure inside the process.

The system of equations is nonlinear (especially due tothe form of �a) and impossible to solve analytically. Ourapproach is therefore to discretize the continuity equation

(Eq. 23) and to iterate the equations on a computer. Com-plete details of the discretization are contained in theAppendix.


Table 3 summarizes the values of the parameters used in theiterative version of the Actin Reconcentration Model thatare not included in the previous two tables. For conve-nience, we choose a diffusion coefficient of D � D� �1.5 � 105 cm2/s for all the salt ions, which lies within therange of values for Na�, K�, and Cl (Stein, 1990).

Monomeric actin is added to the actin filament directlyfrom the profilin-actin complex. This is thought to occur atleast as rapidly as actin polymerization without profilin(Pantaloni and Carlier, 1993; Pring et al., 1992). We there-fore use the in vitro on- and off-rate constants for actinpolymerization: k� � 11.6 (�M � s)1 and k � 1.4 s1

(Pollard, 1986). In the Brownian Ratchet Model, the fila-ment ends grow against an opposing force, in this case, themembrane tension. Peskin et al. (1993) quote a value of � �0.035 dyne/cm for a typical membrane tension, so the loadforce per fiber of f � 2�r�/Nf � 0.1 pN is appropriate forthe acrosomal process. In any case, the final extension rateis limited not by the polymerization rate but by the rate ofactin monomer delivery, so the extension rate is almostentirely insensitive to the choices of k�, k, and f underphysiologically relevant conditions. The temperature T, alsorequired for the Brownian Ratchet Model, is set to T � 300K (room temperature).

The hydraulic permeability Pw of the membrane can beestimated from observations of the doubling of the peria-crosomal region. Because the solute concentrations insideand outside the compartment, the initial and final volumes,and the length of time to change volume (50–70 ms (Inoueand Tilney, 1982)) are all known, it is possible to integrateEq. 25 over time and find a permeability that is consistentwith the experimental observations: Pw � 5.2 � 102 cm/s.That permeability is about seven times the permeability of ared blood cell (Macey and Brahm, 1989), and about one-third the permeability of proximal tubules in rabbit kidneys(Agre et al., 1993). While large, it falls within the range ofpermeabilities exhibited by other biological membranes. Asthe sperm has no other function than to fertilize the egg, it

TABLE 3 Additional parameter values used in iterativemodel


D�, D Ion diffusion coefficients 1.5 � 105 cm2/sk� Actin on-rate constant 11.6/(�M � s)k Actin off rate 1.4/sf Load force per filament 0.1 pNT Temperature 27°C (300 K)Pw Membrane water permeability 5.2 � 102 cm/s� Segment length 0.1 �m�t Time step 0.2 �s


is not implausible that its membrane may be especiallywater-permeable for the purpose of driving elongation ofthe acrosomal process.

The size of the segments over which the equation wasdiscretized was chosen to clearly resolve the structure of theconcentration profiles. We used a segment size of � � 0.1�m length throughout. A time step of �t � 0.2 �s isadequate to maintain stability of the iteration at that segmentsize. Further reduction in the time step or segment size haslittle effect on the results. As before, the initial length of theprocess L0 is set to 0.19 �m. This length extends beyondone segment but does not place the process’s end at aboundary or a center (multiples of � and �/2 are both specialcases for our algorithm; see the Appendix for details).

Results

The model described in this section can calculate resultscorresponding to all of the models in the second and thirdsections. Ion transport can be disabled by setting j-max � 0,and water movement can be prevented by making the mem-brane impermeable to water (Pw � 0). Unless otherwisenoted, the parameters in Tables 1–3 were used in calculatingthe results of the models.

Fig. 8 shows L(t) plots for a variety of parameters andmodels. Each label indicates a pair of curves, one calculatedby the full computer-solved model (dashed lines), and theother by the simpler analytic models (solid lines) from theprevious sections. The solid lines have all appeared inprevious figures, and their labels have not changed. Curve(a) corresponds to the uncorrected model from the secondsection and curve (f) to the corrected model from the same

section. Curve (g�) is the prediction of the model thatincludes nonideality and ion pumping, with a pump rate ofj-max � 3 � 1014 ions/(cm2 � s).

In each case, the predictions of the simpler analytic model(solid line) and the computer-based model (dashed line) lienearly on top of one another. Thus, the models of the secondand third sections semi-quantitatively reproduce the resultsof the full computer model, justifying the approximationsthat we made in those sections, including infinitely fastpolymerization, linear activity profiles, and one-dimen-sional diffusion without bulk fluid flow.

Fig. 9 a shows the concentration of actin in the process atfour times during the extension of the process. These datacome from the computer-solved version of the Actin Re-concentration Model with the parameters in Tables 1–3(corresponding to the dashed line of curve (g�) in theprevious figure). The four profiles correspond to t � 2.5, 5,7.5, and 10 s after the process begins to grow. They are notthe flattened profile with an exponential drop-off one mightexpect for a diffusive system with significant bulk flow,because the overall bulk flow in this system is small. Norare they the linear profiles one would expect between twoendpoints maintained at different concentrations, becausethe driving force for diffusion is the activity of actin, not theconcentration. The concentration profiles shown do, how-ever, translate to activity profiles that are nearly linear (Fig.9 b), providing evidence that our assumption in the secondand third sections of linear activity profiles at all times wasa good one.

DISCUSSION

The question of whether or not diffusion-based actin trans-port is sufficient to support the observed extension speed ofthe Thyone acrosomal process has been answered first in thepositive (Tilney and Kallenbach, 1979) and later in thenegative (Perelson and Coutsias, 1986). Other models havealso been proposed. Oster et al. (1982) suggested that afterthe periacrosomal region doubles in volume, the walls ofthat region, elastically stretched, would push the fluid for-ward, extending the process by hydrostatic pressure. Theactin core would not extend with the process, but catch uplater. The slow rate of actin monomer delivery would there-fore not limit the growth of the process. The calculationsobtain experimental speeds, by fitting some model param-eters to the data, rather than independently determiningthem. Moreover, hydrostatic pressure equalizes at about thespeed of sound in a fluid, and it is therefore unclear why themembrane tension after swelling should be greater at therear of the process than at the front, as would be necessaryto cause elongation. In another paper, Oster and Perelson(1987) suggest that the osmotic movement of water into theacrosome through the membrane at the process’s tip mayassist in extending the acrosomal process by “inflating” itwith fluid. But they provide no plausible driving force forthis influx, and they also note correctly that such an influx

FIGURE 8 Comparison of L(t) predictions of the simpler analyticalmodels of the second and third sections (solid lines) with the predictions ofthe more elaborate computer-based model of the fourth section (dashedlines). Each labeled set of curves contains both a solid and dashed line,even if they are not distinguishable. Curves (a), (f), and (g�) are as in Figs.2, 3, and 5, respectively. In all cases, the predictions of the two methods arenearly identical.


of fluid would interfere with monomer delivery to the tip.Zhu and Skalak (1988) used their model of pseudopodprotrusion to describe the extension of the acrosomal pro-cess. In that model, actin monomers travel by both diffusionand fluid flow, the fluid being driven forward to fill thegrowing tip. Unfortunately, the model parameters are notindependently evaluated, and the central part of the growthcurve is fit by scaling the length and adjusting the zero oftime.

In this model we have tried to incorporate more biolog-ical detail than was present in earlier diffusion-only models,

and we have used only parameters whose values can befound in the literature. While Thyone data may not beavailable for every parameter (e.g., ion pump characteris-tics), the literature contains plausible baseline values forevery parameter from other systems.

Limitations and approximations

In our calculations, we have made a number of approxima-tions that, we believe, simplify the system to a level that ismanageable without losing its essential elements. Nonethe-less, we will examine the approximations and the expectedresults were they to be removed.

Some of the approximations made in the second and thirdsections were drastic: infinitely fast polymerization, ca,tip �0, equilibrium (linear) activity profiles, and infinitely fastwater movement. Often the process was ignored comparedto the periacrosomal region, which has greater volume,surface area, and solute content. As demonstrated by theclose agreement with the previous section’s computer-solved model (which removed those assumptions), theseapproximations are all justified.

In the previous section, when calculating the flow of fluidforward from the periacrosomal region to the tip of theacrosomal process, we simply calculated the volume ofwater that had escaped the extension and required that thesame volume be drawn in from the reservoir to the rear. Thevolume drawn in was spread over the cross-sectional area ofthe extension to give a flux. The fluid flow may in fact bevery different from what we describe. Because the actinfilaments are fairly close together and the concentration ofactin monomers is fairly high, the system may more closelyresemble a set of actin “balls” rattling around in a lattice ofparallel actin “wires” rather than solute in smooth fluidflow. A detailed hydrodynamic calculation (a difficult prop-osition) would account for these effects. However, becausebulk flow plays such a small role in the Actin Reconcen-tration Model, the effect of a more detailed treatment ofhydrodynamic effects is expected to be very small.

We have assumed throughout that diffusion coefficientsdo not vary with solute concentration, when it is known thatself-diffusion coefficients decrease when concentration in-creases (Han and Herzfeld, 1993; Muramatsu and Minton,1988). However, unlike other cells, most of the protein inthe acrosomal process is actin, and therefore mutual diffu-sion is more relevant than self-diffusion. The presence ofsome higher molecular weight proteins and some free pro-filin has been ignored in our models. Although these pro-teins will slow actin diffusion somewhat, they will alsoincrease the solute activity, which is the driving force fordiffusion (Han and Herzfeld, 1994). Thus the decrease inthe diffusion coefficient due to crowding by other solutes isat least partially offset by the increased driving force (ac-tivity) due to the same crowding, and a more rigorousexamination of diffusion in such a concentrated mixture isbeyond the scope of this paper.

FIGURE 9 Plot of the concentration (a) and activity (b) of actin (mM) asa function of position in the process (�m) for the iterative computer versionof the Actin Reconcentration Model. The parameters correspond to thoseused for curve (g�) in Fig. 5. The four curves correspond to snapshots at2.5, 5, 7.5, and 10 s after the beginning of the acrosomal reaction.


Termination of process growth

In the course of trying to answer how the acrosomal processcan reach 60–90 �m in 10 s, it became clear that a secondquestion is also important: why does the process stop grow-ing? Experimentally, the length of the process increasessteeply but then suddenly plateaus. The sharpness of thecrossover suggests a definite mechanism or condition forhalting. By itself, depletion of actin from the system resultsin a gradual decrease in extension speed as the actin con-centration gradually falls.

Depletion of membrane may halt process growth. Onceall of the available membrane in the system has been dis-tributed to cover the process and reservoir, polymerizationat the tip would face increasing resistance, eventually stop-ping when the membrane can no longer deform enough toadd another monomer. (This corresponds to reaching the“stall force” of the Brownian Ratchet Model: the force atwhich dL/dt 3 0 in Eq. 24.)

Effects of osmotic pressure variation

The effects of hyperosmotic and hypoosmotic conditions onthe growth of the acrosomal process of Thyone have beenexperimentally probed by Tilney and Inoue (1985). In brief,they found that hyperosmotic conditions tended to suppressgrowth of the acrosomal process, with no growth at allfound at 150% of normal tonicity. The opposite effect wasalso seen to hold: a decrease in external osmotic pressureincreased the extension rate. A straightforward qualitativeinterpretation of these results is that actin monomer releasein the initial reaction is correlated with the degree of swell-ing of the periacrosomal region. Unfortunately, no detailedmodeling of these experiments can be made without know-ing either the volume of water or the amount of salt thatenters the periacrosomal region under the altered osmoticconditions.

Experimental tests

Thyone sperm is not very amenable to internal manipula-tion, either before or during the acrosomal reaction, makingthe Actin Reconcentration Model difficult to test experi-mentally. However, the predictions of the model depend ontwo properties of the more accessible cell membrane: itsability to transport water and ions. The membrane’s hydrau-lic permeability can be reduced by inhibiting membranewater channels with mercurial sulfhydryl reagents (Agre etal., 1993). However, this would inhibit water influx duringthe initial swelling as well as water efflux following themovement of salt out of the process. Alternatively, inhibit-ing the ion pumps (e.g., by ouabain (Stein, 1990)) wouldprevent salt from being removed from the system and pre-vent the reconcentration of actin in the reservoir. The ex-tension rate would then drop dramatically, falling to diffu-sion-only levels (i.e., from curve g� to curve f� in Fig. 5, aand b). No other models for acrosomal process extension

rely on ion pumps, so any decrease in extension speed seenwhen those pumps are inhibited would be evidence in favorof the Actin Reconcentration Model.

Extension to other systems

In principle, the Actin Reconcentration Model could beapplied to the extension of the acrosomal processes of otherechinoderms. For many such species, the acrosomal reac-tion is similar to that of Thyone (Dan, 1967), although thelengths of the final processes are usually much shorter.However, the data necessary to set system parameters maybe more difficult to find, as the acrosomal process of Thyoneseems to have been studied in more detail than that of otherorganisms.

Other systems with actin-based motility such as goldfishkeratocytes and Listeria monocytogenes do not have thedifficulty that Thyone does with actin delivery. In thosesystems, actin monomers are stored relatively close to thesites where polymerization takes place, so diffusion is fastenough to supply monomers. In addition, none of thosesystems exhibits rapid ion and water movement, so theActin Reconcentration Model is probably not relevant.

CONCLUSIONS

Previous theoretical studies of the extension of Thyone’sacrosomal process have concluded that diffusion cannotdeliver actin to the tip fast enough to support the observedextension speeds. These models also overlooked some im-portant features of the system. The results of our ActinReconcentration Model suggest that, due to the effects ofnonideality and the concentrating effects of ion and waterexport, diffusion is much faster than previously thought.The model quantitatively predicts extension speeds in therange of those observed experimentally using values fromthe literature for all parameters. Both salt transport andsolution nonideality are ubiquitous and well-understoodphenomena. It appears that their concerted action in theacrosomal process of Thyone creates exceptional functionalbehavior.

APPENDIX

Details of the numerical model

This appendix contains the details of the numerical solution described inthe fourth section. The acrosomal process is divided into cylindrical seg-ments of length � and radius r. The segments are indexed by m, with m �0 adjacent to the periacrosomal reservoir and m � mtip at the tip. Theperiacrosomal compartment is labeled as if it were another cylindricalsegment with m� res. Because the overall length L of the process is rarelyan even multiple of �, segment mtip is of variable length: �tip � L mtip�.The full cross-sectional area of the cylindrical process is �r2, but thecross-sectional area accessible for diffusion (unoccupied by filaments) isAd � �r2. The fluid filled volume of segment m is then Vm � Ad� for 0 �

m � mtip. For m � mtip, Vmtip� Ad�tip. The surface area of a segment is

Sm � 2�r� for 0 � m � mtip, and Smtip� 2�r�tip � �r2 for m � mtip.

Because the surface area of the entire system is fixed at S0, the surface area


of the reservoir is the area of the membrane not in the process: Sres � S0 �r2 2�rL.

The concentration of each solute is tracked in each segment as afunction of time. The solutes are denoted by the subscript i � a, �, or ,corresponding to actin and the two types of salts, respectively. ci,m is theconcentration and ai,m is the activity of the ith solute in the mth segment.By integrating Eq. 23 over a time step �t and segment length �, we arriveat the discrete diffusion-convection equation for segments 0 � m� mtip 1:

ci,m�t �t� � ci,m�t�� Di

ai,m�1 � 2ai,m ai,m1

�Ad�t

� ci,m1�Vm13mf ci,m�Vm3m�1

f �i,m� 1Vm

(A1)

where Di is the diffusion coefficient of the ith solute, and �i,m is the numberof particles of the ith species that leave the solution from segment m duringa time step �t (from the i in Eq. 23). The fluid speed � in Eq. 23 has beenrewritten in terms of the volume of fluid �Vm3m�1

f that moves fromsegment m to segment m � 1 in a time step �t.

The volume of fluid that is drawn from segment m into segment m � 1,�Vm3m�1

f , includes the volume needed to fill the extending process andthe sum of all the volume leaving segments further forward than segmentm. The volume needed to fill the process is a function of the extensionspeed: �extAd�t. The volume of fluid that moves through the membranedepends on the osmotic pressure difference across the membrane. Ex-pressed as a concentration difference, the osmotic driving force in segmentm is (see Eq. 26)

��m � �ext � c�,m � c,m, (A2)

where �ext is the external osmotic pressure. Equations 25 and A2 togetherimply that the volume of water �Vm

� that will cross the membrane area Sm

in a time �t under a driving force ��m is

�Vm� � �wPwSm�t��m, (A3)

where Pw is the hydraulic permeability of the membrane, and �w is thevolume of a water molecule (to convert from the particle flux given by theother terms in the equation to a volume flux). The total fluid volume thatleaves segment m is therefore

�Vm3m�1f � �extAd�t �

n�m�1

mtip

�Vn�. (A4)

The sink terms �i,m are straightforward. The �-salt ions experience nonet transport in or out of the system, so ��,m � 0 everywhere. Actin is onlypolymerized at the tip, so �a,m � 0 for m � mtip. For m � mtip,

�a,mtip � �Nf/��ext�t, (A5)

where Nf is the total number of actin filaments in the process, and �ext is theextension speed of the process. For the -salt there is net transport. UsingEq. 13,

�,m � j-max

c,m

c,m c,1/2Sm�t, (A6)

where the sign explicitly indicates a movement of particles out of theprocess. A similar equation for the -salt ions applies in the reservoir.

The iteration begins just after the beginning of the acrosomal reaction.The initial influx of ions and the swelling of the periacrosomal region arenot modeled in detail. Instead, the initial volume of the reservoir is set tothe doubled value (Vres(0) � 2V0), and the initial concentrations of actinand the ions are set to their post-swelling values: ci,res(0) � Ni0/(2V0)(i �a, �, or ). The initial length of the process L0 is chosen so that there are

initially two segments, and the concentration of actin in these segments isinitially the same as in the reservoir.

Once each time step, the solute concentrations in the process andreservoir are updated using Eq. A1. Each update begins with actin poly-merization at the process tip. The extension speed can be calculated fromthe polymerization rate. Then the new concentrations are calculated, be-ginning in the tip segment. The solute concentrations in the tip segmentchange not only due to diffusion and bulk fluid flow, but also due topolymerization. Furthermore, extension of the process changes the size ofthe tip segment. After the tip segment is updated completely, the concen-trations in each segment in the rest of the process are updated, workingfront to back. Finally, the concentrations in the reservoir are updated. Thedetails of these steps are described in the remainder of this section.

The extension speed �ext is given by the Brownian Ratchet Model (Eq.24). The relevant activity aa is in principle the activity of actin at the tip,aa,mtip

. However, the length of segment mtip is �0.1 �m: probably largerthan the region in which polymerization takes place. So in order to betterrepresent the actin concentration profile near the tip without using adrastically smaller segment size, the concentration of actin is extrapolatedto the forward end of segment mtip:

ca,end �� 2�tip

� �tipca,mtip �

�tip

� �tipca,mtip1. (A7)

The extension speed is then found by using Eq. 24 (calculating the activityfrom the concentration in Eq. A7 via Eq. 18). The new process length isthen:

L�t �t� � L�t� �ext�t. (A8)

The solute concentrations in segment mtip are then updated using therelevant variation of Eq. A1:

ci,mtip�t �t� � ci,mtip�t� �Di

ai,mtip1 � ai,mtip

�tip/2 �/2Ad�t

� ci,mtip1�Vmtip13mtip

f �i,mtip� 1Vmtip

. (A9)

In Eq. A9, the denominator of the diffusion term has been adjusted toaccount for the nonstandard length of segment mtip.

Because the right-hand side of Eq. A9 is evaluated at time step t, thenew tip concentrations at time t � �t have been calculated for the volumeof the old tip segment. The process, however, has grown, and therefore thelength of segment mtip must be updated to reflect the new overall length ofthe process. It becomes ��tip � �tip � �ext�t, and the concentrations of thesolutes in segment mtip are adjusted so that matter is conserved:

c�i,mtip � ci,mtip

Vmtip

Ad��tip. (A10)

Clearly segment mtip cannot grow indefinitely. Although it is mostnatural to allow it to grow until it reaches length � and then to add anothersegment (labeled mtip � 1), the procedure has a practical drawback. If theprocess were to extend only a small distance into the newly added segment,the volume of that segment would be much smaller than the typicalsegment, and the calculation could become unstable. To avoid that situa-tion, segment mtip is allow to vary in length in the range � � �tip � 3�/2rather than 0 � �tip � �. When the segment becomes longer than 3�/2, itis split into two segments, one of length � and one of the remaining length,L (mtip � 1)� � �/2. In this way, the segment does not become verysmall, and the calculation remains stable. The concentrations in the twosegments after the split must again be adjusted. They are set so that allsolutes are conserved, and so the extrapolated actin concentration of Eq.


A7 does not change:

c�a,mtip�1 ��tip 2�

��tip �ca,mtip �

�

��tip �ca,mtip1

c�a,mtip � �ca,mtip�Ad � c�a,mtip�1Vmtip�1�1Ad�

,

(A11)

where the primed concentrations are those after segment mtip is split intosegments mtip and mtip � 1.

After the update step in segment mtip, segments m � mtip 1 to m �0 are updated using Eq. A1, looping backward from the tip to the reservoir.Then the reservoir is updated. Because the reservoir becomes depleted ofsolutes and fluid volume, the absolute number of solute molecules and thevolume of the reservoir are calculated instead of the solute concentrations.The reservoir loses all of the volume that has moved into the process andthrough the process membrane as well as volume through its ownmembrane:

Vres�t �t� � Vres�t� � �Vres30f � �Vres

� . (A12)

Because the reservoir has only one neighboring segment, the updateequation is

Ni,res�t �t�

� Ni,res�t� �Di

ai,0 � ai,res

�� ci,res�Vres30

f �Ad�t �i,res,

(A13)

where Ni,res is the number of molecules of the appropriate solute in thereservoir. The solute concentrations in the reservoir are then simply

ci,res�t �t� �Ni,res�t �t�Vres�t �t� . (A14)

This work was supported by the National Science Foundation under GrantHRD-9021929 and the National Institutes of Health under Grant HL-36546.

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An Analysis of Actin Delivery in the Acrosomal Process of Thyone

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